Integration by Substitution and by Parts

Integral calculus assists in the determination of any given function’s antiderivatives. Such antiderivatives are often known as function integrals. Integration refers to the process of determining a given function’s antiderivative. Determining integrals is just the opposite process of determining derivatives. Any function’s integral depicts a family of curves. The basic calculus consists of determining both derivatives as well as integrals.

Definition of Integral Calculus

Integrals represent the values of the function determined during the integration process. Integration is the method of obtaining f(x) from f'(x). Integrals provide numbers to functions to express displacement and motion questions, area and volume problems and other problems that come from combining all of the little data. We may derive the function f using the derivative f’ of the function f. The function f is the antiderivative and integral of f’.

Different Types of Integrals

Integral calculus is used to solve the following categories of issues:

• the difficulty of determining the derivative of a function, and

• the challenge of determining the region enclosed by a function’s graph under certain circumstances.

As a result, integral calculus is classified into two categories:

• Definite Integrals

• Indefinite Integrals

Definite Integrals

A definite integral describes a number that produces a constant solution. A definite integral has an upper and lower limit at all times. The definite integrals’ boundaries are constant. A definite integral is often defined as an indefinite integral assessed across lower and upper bounds. Because the value or solution produced by calculating the integrals using limits is constant, these integrals are definite. The outcome might be either favourable or bad. The answer to a specified integral is always in a particular region. When evaluating a function with two limits, a definite integral is utilised.

Indefinite Integrals

An indefinite integral is an integral having no constraints. The indefinite integral represents a family of functions with derivative f. In the case of an indefinite integral, there are no bounds.

The answer found by calculating the undetermined function of an indefinite integral is a generalised solution with variables. The area of an indefinite integral’s solution is not given. Indefinite integrals are employed when a broad solution to a problem is needed.

What are Integration Techniques?

For obtaining the integral of complicated functions, we use several integration techniques. To simplify integral issues, we must first determine the kind of function to be integrated and subsequently use the integration procedure. In order to minimise the trigonometric functions within integration, we also employ trigonometric formulae and identities as an integration technique.

Techniques of Determining Integrals

The standard integration techniques are only applicable if the integrand is in the ‘standard’ form. Most applications or evaluations demand specialised integration methods that reduce the integrand theoretically to a solvable form. The most common approaches are –

• Substitution method

• Integration by parts

Substitution Method

The substitution technique of integration is also known as the u-substitution technique. To simplify the function, we may use this way to modify the variable of integration. It is identical to the reverse chain rule.

For example, consider integration of the form ∫g(f(x))dx.

Then, assuming f(x) = u, we may replace f(x).

This further implies that,

f,(x)dx=du

dx=du/h(u)

Where,

h(u)=f,(x)

Please keep in mind that if we modify the variable of integration, we must modify it throughout the integral. As a result, the substitution technique integration formula is:

∫g(f(x))dx=∫g(u)/h(u)du

Integration By Parts

The integrand under this approach is often a product of two functions (whose integration formula is known beforehand). Either of the functions is the ‘first function,’ while the other is the ‘second function.’ The fundamental formula underlying the integration by parts approach is:

∫(uv)dx=u∫v.dx-∫[u,∫v.dx]dx+ c

Where u and v are functions of x.

u(x)= 1st function 

v(x)= 2nd function

Assume your integrand is a combination of two functions: exponential and logarithmic. For ease of evaluation, the logarithmic function would be selected as the first function, and thus the exponential function would be selected as the second function

Conclusion

We can determine the distance from the velocity by integrating. Definite integrals are a useful tool for calculating the area under simple curves, the area bounded by a curve and a line, the area between two curves, and the volume of solids. Integrals are often used to solve displacement and motion questions. The functions to integrate may be broken into a sum or difference of functions, where the integrals are known individually. After obtaining the indefinite integral of the function, always add the constant of integration.